Mathematical models of the solid cancer induced by atomic bomb and the spontaneous cancer in the daily life—Proposal of a new medical treatment for cancers

Abstract Background A mathematical model of the radiation−induced cancer was devised to explain the change of incidence rates pursued by Radiation Effect Research Foundation for 25 years. Aim The aim of this work is construction of mechanisms of radiation‐induced cancer and cancers observed in the daily life. Methods and results First, we found a way to separate spontaneous cancers from radiation‐induced cancers observed among atomic‐bomb victims in Hiroshima and Nagasaki districts by using a constructed algorithm. The isolated incidence rates of radiation‐induced cancers were reproduced by a two‐stage model mechanical collision of impinging radiation with cells and succeeding mutation of the damaged cell to cancer. This model satisfactorily reproduced observed solid cancer incidence rates. We further attempted to construct a mathematical model for the age‐dependence of spontaneous cancers appearing in the daily life and concluded that the cancer should be generated at cell division. Conclusion With these findings, we reached to a conclusion that cancers may be suppressed by eliminating damaged cells with mild–dose radiation.


| INTRODUCTION
Fukushima-Daiichi nuclear power plant accident in 2011 evoked a great interest on the exposure, particularly internal exposure among the people living in Japan at the moment. However, the object of discussion was concentrated on the allowable exposure level and any quantitative discussion on the radiation injury was never attempted.
The typical injury caused by radiation is the cancer, particularly that observed among HiroshimaÀNagasaki atomicÀbomb survivors.
Solid cancers except for cancer of blood, malignant tumor etc. were energetically followed up by a group 1 mainly consisting of the members of Radiation Effects Research Foundation (RERF) from 1975 to 1999. This valuable report is the only one quantitative document on the incidence rates for solid cancers among HiroshimaÀNagasaki atomic bomb survivors.
Several investigations 2 are carried out using the RERF data. However, their interests were concentrated on the comparison of radiation risks among cohorts with respect to sex, age at the time of the bombing, radiation dose or dose rate, and so forth. In such a study, it is conceivable that the fundamental part of the mechanism of carcinogenesis has been washed out. None of the proposed models succeeded in giving quantitative explanation for the RERF data. 1 Therefore, we may have to look for a different approach to the carcinogenesis.
In 1957, Armitage and Doll published a multistage model (AD model) 3  Then, we attempted reproducing the incidence rates reported by RERF gropu. 1 Solid cancers caused by radiation are clinically indistinguishable from cancers naturally occurring in the daily life. The incidences given in reference 1 contain those of the spontaneous cancers. Therefore, we first built a mathematical model for reported incidence rates containing the spontaneous cancer and devised a method deducing the magnitude of the spontaneous cancer by means of thus derived formula for the model.
Thus, we could estimate the net incidence rate for the radiationinduced cancer. Then, the above radiochemical cancer model was applied to the net incidence data.
Finally, we attended to the age-dependence of spontaneous cancer incidence rate to make a model of spontaneous cancer model. We noticed the nearly exponential distribution of the death rate by cancer versus age and concluded that the mutation takes place at the moment of cell division. With thus constructed whole story of cancers, we found the possibility of a harmless medical treatment of cancers using low-level radiation.

| RISK OF SOLID CANCERS
Sudden mutation was shown by Manabe et al. to explain with a simple reaction rate formula 12,13 which does not require consideration of existence of the latent period before mutation takes place. On the contrary, solid cancers induced by radiation requires quite long latent periods. To reproduce such latent periods, the model needs to be divided into two steps, the process giving damage to cells by radiation exposure and the transformation process of damaged cells to cancer cells.
Mechanism of the two-stage radiation-induced cancer 4-11 consists of two fundamental radiochemical processes, mechanical collision between an impinging neutron, electron or photon and electrons bound to a chemical bond to produce unstable fragments, and a statistical event equivalent to the decay of radioactive nuclides.
This principle was applied to the research data 1 on the incidence of solid cancers among HiroshimaÀNagasaki survivors to construct a mathematical model. The first step producing radiation damage is a simple physical process and can be treated essentially likewise as the reaction rate formula of sudden mutation1. 12,13 The second step for the latent period is a statistical process following the first-order reaction, and can be treated as the radioactive decay, that is, the growth rate equation for the daughter nuclide in the radioactive decay. The number of damaged cells is given by γD N 0, Then the change of damaged cells is written as Here, α and β are reaction rate constants while γD is the rate of damaged cells produced at the moment of bombing to the total cells. The first term of the right side of Equation (1) is the number of normal cells at time t after bombing. The second term represents the relaxing process of damaged condition by radiation and β gives the rate constant of relaxation process.
Equation (1) is transformed for one normal cell as follows where N = N d /N 0 . The solution of Equation (2) is where μ ¼ α þ β and f ¼ α=μ The continuous time variable is now replaced with an integer n corresponding to 1 year. That is, 1 year is regarded to be the unit of time. The factor for the latent period is introduced into the equation where N c (n) is the number of cancer cells at the nth year after bombing and Λ is the rate constant of mutation.
The incidence rate is given as the number of patients per 10 5 populations, so that we must replace the concept of the number of damaged cell by the number of patients. This procedure may be allowed because the number of persons received multiple damages is expected to be negligibly small considering a quite low geometrical cross section for contact of radiation.
We can set a condition Λ < μ from the characteristics requested to Λ and Equation (4) is transformed as follows; 1 À e Àμ Á n 1 À e Àμ À e ÀΛ Á n À e Àμ Á n 1 À e À μÀΛ Here, B is a secondary parameter like μ and f defined by As is obviously seen, the incidence rate N c is expressed with four parameters f, μ, Λ, and B. However, f is a scaling factor and not an independent parameter excluded in the fitting procedure. The two parameters B and γD obey the restrictions that B > À1 and γD < 1, respectively.
The incidence rates given in the RERF report 1 are the sum of people affected by cancers for 5 years. Therefore, Equation (5) is written as to compare the model with observed data.

| INCIDENCE OF STOMACH CANCER
Among 33 cancers given in reference 1, the incidence rate of cancers of stomach (for Male and Female), cervix uteri and gallbladder (for Female) are decreasing after 30 years since bombing. This trend was never reproduced by changing values of above three free parameters.
This implies that we need to introduce another function guaranteeing decrease of the incidence with time.
In 2001, a report was published that no stomach cancer patients appeared among people not infected with pylori bacteria, 14,15 while occurrence of stomach cancer was observed among people infected with the bacteria. 16 Suppression of cancers may be attributed to the immunity function of removing bacteria. Decrease of incidence is also observed among gallbladder, cervix uteri, and so forth. 1 Therefore, we must find a universal type of immunity function for incidence.
Recently Destruction of antiÀimmune cells by radiation is the firstÀorder reaction and controlled by the same equation for the decay of radioactive nuclides. Then, the incidence risk of stomach cancer at n years after bombing is given by Then, the incidences for 5 years is given by We first surveyed the structure of the χ 2 values in the phase space constructed with three parameters, μ, Λ, and B, before starting the search of the bestÀfit condition, where χ 2 is a measure of deviation of the calculated value from the observed one. The vest fit is obtained by a minimum spot of χ 2 in the phase space of the above three parameters. The minimum spot is searched T A B L E 1 Observed and calculated incidence rate (number of incidence per 10 5 population) during 5 years after the bombing for Hiroshima-Nagasaki atomic bomb survivors (Male) by the least squares method with a computer. However, obtained twoÀdimensional contour map of χ 2 revealed very shallow valley without any clear minimum spot as shown in Figure 1 for stomach cancer. Therefore, the least square fitting procedure using contour map of χ 2 does not work for the present case because the final answer primarily depends on initial input data and does not necessarily merge to the true destination.
However, we can introduce another severe condition that μ and Λ must stay, respectively, at a common value, respectively, throughout reported 33 cancers, because the processes described by Equations (1), (2), and (8)  condition until somewhat satisfied fitness was obtained as shown in Tables 1 and 2, together with deduced values of the parameters in Table 3. Table 3 indicates the 33 cancers are divided into two groups.
Majority of them satisfy our speculation, while the remain minor group gathered with a set of slightly smaller common values of the three parameters, μ, Λ, and B. The reason why they are separated to two groups are not clear.

| ESTIMATION OF SPONTANEOUS INCIDENCES
Observed incidence rates in Tables 1 and 2 include spontaneous incident rates induced with no radiation effect as stated before. Unfortunately, we are unable to clinically distinguish the radiationÀinduced and spontaneous cancers. Mixing of the spontaneous incidence might give us a false view of the mechanism of cancer.
T A B L E 3 Parameters for solid cancers including spontaneous cancers There are two difficulties for searching spontaneous incidence rates among published data. The first problem is that it is practically impossible to know the exact incidence time for spontaneous cancers among general people contrary to carefully traced inhabitants in Hir-oshimaÀNagasaki area. The second difficulty is the difference in the age composition of population. The peak of age composition of population in Hiroshima and Nagasaki is considered to shift toward younger age compared to the national population because aged people passed away to a greater extent due to their physical weakness under such a severe circumstance. The death rates in reference 18 stays at a very low level under 50 years old but rapidly increases with age thereafter. This would result in smaller incidence rates (death rates) under the influence of radiation. Spontaneous incidence rates under the normal condition were estimated using total number of incidence by the spontaneous cancer 18  The method to subtract the spontaneous incidence rates from the radiationÀinduced data must be found as a statistical way if possible. First, let us transform Equation (9) by introducing a minor approximation of changing the exponential factor e Àε Á j to e Àε Á n and taking the exponential factor out of the summation with respect to j. We further introduce a free parameter Q sc as the spontaneous incidence rate and a physical quantity ξ(n) defined by These procedure leads to an equation; Equation (12) now requires that ln ξ (n) should show a linear dependence with respect to time. That is, the spontaneous incidence rate Q sc is determined so as to satisfying Equation (12)  Now, we return to Equation (9) and try to find the best choice of free parameters so as to reproduce the net incidence rate, {Q obs (n) -Q sc } which are given in Tables 4 and 5, together with deduced spontaneous incidence rates Q sc . Temporarily obtained values of parameters for the whole incidence rate in Tables 4 and 5 were considered as the initial guess in the fitting procedure for the incidence rates of net radiationÀinduced cancers given at the upper line for each cancer in Tables 4 and 5.
After a steady and persistent effort, sufficient reproduction was achieved as shown in Figure 3A-C and deduced values of the parameters were summarized in T A B L E 5 Observed and calculated incidence rate (number of incidence per 10 5 populations) of net radiation-induced solid cancers (second group) F I G U R E 3 Observed and calculated incidence rates of net radiation-induced solid cancers during 5 years for Hiroshima -Nagasaki atomicbomb survivors cancers. It is more serious problem to formulate the mechanism of spontaneous cancers, contrary to the former induced only in the emergency. However, any systematic yearly change in the incidence is not expected to appear among patients of spontaneous cancer contrary to the radiationÀinduced cancers. So, we must look for other sources of data on which we construct a mathematical model.
The risk of a cancer starts at the moment of birth and people keep staying under the risk thereafter. The age therefore corresponds to the exposed duration of HiroshimaÀNagasaki survivors. The data of yearly cancer incidence rates of age groups for 5 years given by National cancer center, Japan 20 are our subjects to try to construct the mathematical model of the spontaneous cancer.
The devised model is based on the abnormal cell division. We adopted the yearly incidence rates 20 of age groups for investigating the mechanism of spontaneous solid cancers. Age-dependent features of incidence rates reveal quite different aspects from the time dependence of radiationÀinduced cancers. The incidence rate stays at very low level below middle age and begin to increase quite violently thereafter. 18 This lets us think of increase of the number of cells in the cell division and increase of abnormal cells produced at division and led to cancers.
However, the increasing rate of the number of cells is turned out too large to reproduce the incidence rate when we compute the number of cells with the condition that two cells are produced at each where ζ is the probability for occurrence of abnormal division at cell The discrepancy in Equation (13) should be attributed to either κ or η. As a result of examination, we reached a conclusion that only reduction of η for high generation can cover the discrepancy.
It is then assumed that η stays constant until it starts to linearly decrease at the age n 0 in the aging effect as shown in Figure 4A,B, F I G U R E 4 Age-dependence of the number of cells survived at cell division Equation (14) for n ≤ n 1, where ϕ is a function defined as for n > n 1 . The values of n 0 , n 1, η 0 , and η 1 are given in Table 7.
The recursion work with age-dependent incidence rates averaged over 5 years from 1995 to 1999 20 was repeated until getting a selfÀconsistent solution for the introduced parameters for an appropriately given κ value. The best result was obtained when we chose the condition κ = 0.9 (13 months as the time between the adjacent divisions), respectively, and the value of η 0 and the resulting functional form of η for each cancer is shown as a combination of a constant part T A B L E 7 Deduced parameters from incidence data of spontaneous cancers and linearly decreasing section above a critical age n 0 as given in Figure 4A,B.
The resulting incidence rates are shown in Figure 5A-E in comparison with the observed values, and numerical data are given in Table 8. The deduced values for the introduced parameters are summarized in Table 7. All cancers were found controlled with two The most noteworthy item in Table 6 is the result that μ and Λ   it is unable to deprive the function of apoptosis. Mutation from the unstable prodromal cell takes place during the latent period, which is generally observed among radiation-induced cancers. These observations unify the two apparently independent models to one framework.
The character of mutation in the second stage of radiationÀinduced cancer corresponds to the abnormal cell division of spontaneous cancers and the parameter β corresponds to ζ of spontaneous cancer.
The former is larger than the latter by two to three figures. This indicates high enhancement of cancer incidence by lowÀlevel radiation.
This model proceeds to an idea that apoptosis is the very essential function for the proper growth of creature and continual existence of creature itself is threatened if this function has ceased to work. The part of apoptosis occupying in the evolution of creation is then realized to be far more important than being imagined so far.
The expected hormesis effect suggests that the possible preventive treatment of lowÀdose irradiation before generation of cancers may be effective for at least half of the observed 37 cancers possessing large ε values given in Table 6. Moreover, the hormesis effect is considered to be a promising measure of suppressing metastasis for all solid cancers.
Medical treatment of lowÀdose irradiation for whole body would be effective for killing damaged cells at pre-cancer stage. Fortunately, the hormesis effect is particularly strong for cancers of stomach, lung, liver, and so forth, showing high incidence rates. Therefore, the incidence rate would be suppressed to a large extent, when this treatment is accepted in a nationwide scale.
When this treatment is proved effective, patients once suffered by cancer will be relieved from the fear of recurrence of cancer without any severe side effect. Recently, it is reported that there are females who cut off their sound breast to avoid the future hazard of breast cancer.
Female adults will certainly be liberated from such a bitter judgment.
If the present model is correct, the proposed medical treatment with lowÀdose radiation is expected to be applicable to other cancers, T A B L E 8